# The nature of the steady state in models of optimal growth under uncertainty

## Author Info

• Tapan Mitra

()

• Luigi Montrucchio

()

• Fabio Privileggi

()

## Abstract

We study a one-sector stochastic optimal growth model with a representative agent. Utility is logarithmic and the production function is of the Cobb-Douglas form with capital exponent $\alpha$ . Production is affected by a multiplicative shock taking one of two values with positive probabilities p and 1-p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters $\alpha$ and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever $\alpha < 1/2$ . More delicate is the case $\alpha > 1/2$ . Singularity with respect to Lebesgue measure also appears for values $\alpha ,p$ such that $\alpha < p^{p}\left( 1-p\right)^{\left( 1-p\right) }$ . For $\alpha > p^{p}\left( 1-p\right) ^{\left( 1-p\right) }$ and $1/3\leq p\leq 2/3,$ Peres and Solomyak (1998) have shown that the distribution is a.e. absolutely continuous. Characterization of the invariant distribution in the remaining cases is still an open question. The entire analysis is summarized through a bifurcation diagram, drawn in terms of pairs $\left( \alpha ,p\right)$ . Copyright Springer-Verlag Berlin/Heidelberg 2003

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL: http://hdl.handle.net/10.1007/s00199-002-0340-5

As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

## Bibliographic Info

Article provided by Springer in its journal Economic Theory.

Volume (Year): 23 (2003)
Issue (Month): 1 (December)
Pages: 39-71

as
in new window

## References

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

as in new window
1. Hopenhayn, Hugo A & Prescott, Edward C, 1992. "Stochastic Monotonicity and Stationary Distributions for Dynamic Economies," Econometrica, Econometric Society, vol. 60(6), pages 1387-406, November.
2. Mirman, Leonard J, 1972. "On the Existence of Steady State Measures for One Sector Growth Models with Uncertain Technology," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 13(2), pages 271-86, June.
3. Brock, William A. & Mirman, Leonard J., 1972. "Optimal economic growth and uncertainty: The discounted case," Journal of Economic Theory, Elsevier, vol. 4(3), pages 479-513, June.
4. Donaldson, John B. & Mehra, Rajnish, 1983. "Stochastic growth with correlated production shocks," Journal of Economic Theory, Elsevier, vol. 29(2), pages 282-312, April.
5. Mirman, Leonard J., 1973. "The steady state behavior of a class of one sector growth models with uncertain technology," Journal of Economic Theory, Elsevier, vol. 6(3), pages 219-242, June.
6. Mirman, Leonard J. & Zilcha, Itzhak, 1975. "On optimal growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 11(3), pages 329-339, December.
7. Brock, W. A. & Majumdar, M., 1978. "Global asymptotic stability results for multisector models of optimal growth under uncertainty when future utilities are discounted," Journal of Economic Theory, Elsevier, vol. 18(2), pages 225-243, August.
8. Futia, Carl A, 1982. "Invariant Distributions and the Limiting Behavior of Markovian Economic Models," Econometrica, Econometric Society, vol. 50(2), pages 377-408, March.
Full references (including those not matched with items on IDEAS)

## Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

## Corrections

When requesting a correction, please mention this item's handle: RePEc:spr:joecth:v:23:y:2003:i:1:p:39-71. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla)

or (Christopher F Baum)

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.